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Abstract: How much rope does it take to tie a knot? And what do the ropelength minimizing configurations of a knot look like?
The answer to these questions is of interest in molecular biology, where the speed of knotted strands of DNA moving through a gel is linked to the minimum ropelength of their knot types.
In this talk, we'll present a survey of some recent results on ropelength-minimizing knots and links, including joint work with Rob Kusner, John Sullivan, Heather Johnston, and Joe Fu.
We'll prove that ropelength minimizers exist and are C1,1 in every knot and link type. We'll discuss "large-scale" bounds on ropelength in terms of crossing number, showing that for every knot,
and constructing knots and links which show that the exponent of the lower bound is sharp. And we'll discuss a new family of ropelength bounds which allow us to construct a family of tight links, and to improve the best known lower bounds on the ropelength of a nontrivial knot.
This talk will be accessible to graduate students.
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